REPORT ON CERTAIN BRANCHES OF ANALYSIS, 259 



C D' and C D will be symmetrical by pairs ; but one portion 

 only, C D, will necessarily coincide with the primitive curve. 



The theory of discontinuous functions has recently received 

 considerable additions from a young analyst of the highest pro- 

 mise, Mr. Murphy, of Caius College, Cambridge. In an admi- 

 rable memoir on the Inverse Method of Definite Integrals *, he 

 has given general methods for representing discontinuous func- 

 tions, of one or a greater number of breaks, by means which are 

 more directly applicable to the circumstances under which they 

 present themselves in physical problems than those which have 

 been proposed by Fourier, Poisson, and Libri. Mr. Murphy 

 had already, in a previous memoir f, given a most remarkable 

 extension to the theory of the application of Lagrange's theo- 

 rem to the expression of the least root of an equation, which 

 we shall have occasion to notice hereafter ; and he has shown 

 that if (p (x) be an integral function of x then the coefficient of 



— in the developement of — log — will represent tlie least 



root of the equation <p x = 0. We thus find that the least 

 of the two quantities « and /3 will be represented by the coeffi- 



cient of — m the series for log ^ ^ ^ ^% which is 



X ° X 



and if we replace « and /3 by — and -r-, the least of the two 



quantities — and -^, or the greatest of the tv/o quantities a. and 

 ^, will be represented by 



1 1 1 — "^ - 113 -(«^'- 



T-^ + ^^ ■ {^y ^ .-^6 • {^^y 



* Transactions of the Philosophical Society of Cambridge, vol. iv. p. 374. 



t Ibid. p. 125. 



J If we represent the series (2.) by S, we shall get 



'^"-'S " =-LorO, 



fjg 



according a^s et is greater or less than /3 : thus — — would represent the at- 

 traction within and without a spherical shell, which is or —3, where « is 



the distance from the centre. 



s2 



